Variational problems with pointwise constraints and degeneration in variable domains

Abstract

In this article we deal with a sequence of functionals defined on weighted Sobolev spaces. The spaces are associated with a sequence of domains Ω s contained in a bounded domain Ω of R n . The main structural components of the functionals are integral functionals whose integrands satisfy a growth and coercivity condition with a weight and additional terms ψs ∈ L 1 (Ωs). For the given functionals we consider variational problems with sets of constraints for functions v of the kind h(x,v(x)) 0a .e. inΩs ,w hereh : Ω ×R → R. We establish conditions on h and ψs and on the given domains, weighted spaces and functionals under which solutions of the variational problems under consideration converge in a certain sense to a solution of a limit variational problem with the set of constraints defined by the same function h.